False Conjecture

(Description of Art)

Let’s link each piece of math-art to its card number. This will be completed by October.

1. The Hat Monotile

The “hat” monotile can tile the plane, but only aperiodically. This means it lacks translational symmetry—you cannot shift the overall pattern in any direction and have it perfectly overlap its original footprint.

2. The Binary Tree

This tree branches in two, 10 times in total. If you look closely with a magnifying glass, you’ll find exactly 1024 little buds on the ends.

3. The Bent Polyomino

The L-tromino (a bent shape made of 3 squares) is the smallest polyomino capable of forming an irreducible rectangle. Specifically, it can tile a 5×9 rectangle in such a way that the overall shape cannot be subdivided into any smaller rectangles made of those same tiles.

4. The Four-Color Theorem

The map of Ukraine can be colored using just four colors so that no two regions sharing a border share the same color. Thanks to the Four-Color Theorem, this is true for all 2D maps, not just Ukraine.

5. The 5-Question Venn Diagram

The 5-question Venn diagram records dice rolls that answer five questions:

• Do some of the dice add to 7?
• Do some of the dice add to 8?
• Do some of the dice add to 9?
• Do some of the dice add to 10?
• Are all the dice different?

One of the regions in this Venn diagram is impossible to fill with a dice roll.

6. Picasso’s Canvas

Picasso’s Canvas: An acceptable paint stroke never completely covers a 2×2 square. Also, the last number visited must be the product of the first number and the total number of squares in the paint stroke. This specific path leaves only 6 unpainted squares, the lowest yet discovered for an 8×8 grid.

7. Mondrian Art Puzzle

Mondrian Art Puzzle: An acceptable canvas is covered by rectangles, no two of which share the exact same dimensions. The score is the difference in area between the largest and smallest rectangles. This solution scores 7, the minimum possible for a 12×12 grid. It remains an open problem whether any sufficiently large square can score a perfect zero.

8. Archimedean Tilings

There are exactly 8 semi-regular (or Archimedean) tessellations: repeating wallpaper patterns made of two or more types of regular polygons where every single vertex is identical.

9. Roll Reversal

This puzzle by Ali Muniz is called Roll Reversal. Start with a 2-7 column and try to make a 7-2 column in the fewest steps. Each step allows you to cut a rod into smaller integer lengths or glue two neighboring rods together. Duplicate lengths are not permitted in a column. Also, at no point can you exceed the length of your longest starting rod.

10. The Lazy Architect

The Lazy Architect: Build a mansion with one room of area 1, one room of area 2… all the way up to one room of area 10. Do this with the minimal number of walls. This particular solution uses 7 horizontal and 7 vertical walls (14 total), but 14 walls isn’t lazy enough. This is not the fewest number of walls possible for the 1–10 mansion.

11. The Fraction Loop

It takes exactly 62 steps to start at 1 and return to 1 using only two allowed operations: adding 1/11 or taking the reciprocal (flipping the fraction).

12. Hexiamonds and Pentominoes

There are exactly 12 ways to glue six equilateral triangles together edge-to-edge (forming shapes known as hexiamonds). Coincidentally, there are also exactly 12 ways to glue five squares together edge-to-edge (forming pentominoes).

13. The Archimedean Solids

There are exactly 13 Archimedean solids. These highly symmetric polyhedra are composed of regular polygons and are “vertex-transitive”—meaning the geometric arrangement of faces around every single vertex is identical.

14. Dissecting the Square

The fewest number of acute triangles needed to perfectly tile a square is 8. (Your original note was a fragment, so I completed the thought! Any dissection using 7 or fewer triangles inevitably forces at least one right or obtuse angle).

15. Banyan Root Puzzle

Banyan Root Puzzle: Starting at 1, add integers sequentially so that each new number is the sum of exactly two previous numbers on the path back to 1. You must always place the new integer in a way that minimizes its distance (steps) from 1 without reorganizing the previously built sequence. Under these strict rules, 15 is the first integer that can be reached at two different distances from 1.